Nested Newton´s method for ICA and post factor analysis

Two distinct topics are dealt with. First, a new method for independent component analysis (ICA) has been constructed that exploits the invariance of criteria under component-wise scaling, which is intrinsic to ICA. This practical and simple ICA method is called the nested Newton\´s method. When the number of the channel of observation is less than a certain level, factor analysis (FA) is ineffective (bound for FA). The target of this paper is these cases. Three of many concrete advantages of the nested Newton\´s method are addressed. i) It is robust against Gaussian noise and outperforms existing methods, such as JADE and Fast ICA, especially under Gaussian noise conditions. ii) It is highly stable globally. iii) Each step resolves itself into two-dimensional (2-D) matrix problems. There is thus no need to deal with gigantic matrices, which means that fewer computational resources are required. Second, a method called "post factor analysis (post-FA)" is described that is aimed to be useful as post-processing for ICA. Although it is functionally similar to conventional FA, post-FA is a completely new method and is more powerful than conventional FA in compensation for its stronger assumption that there are mutually independent sources behind observations. By fully making use of this assumption, post-FA is capable of estimating the noise variance beyond the known limit for FA. Furthermore, it improves the accuracy of ICA to a considerable extent. Any ICA algorithm without prewhitening (pre-WH) or pre-factor-analysis (pre-FA) can be used for preprocessing, although the nested method is a good candidate.